Lab 4 Sample Report
Part 1: Data
1a: Roasting
Total roasting time: 130 seconds (very fast!)
| Time (s) | Temperature (°C) | Energy (kWh) |
|---|---|---|
| 0 | 23.0 | 0.00 |
| 30 | 45.0 | 0.01 |
| 60 | 56.9 | 0.02 |
| 90 | 69.7 | 0.02 |
| 120 | 89.2 | 0.02 |
1b: Grinding
Time: 20 seconds Power: 17.3 W
\(Energy = 17.3 \times 20 \times \frac{1}{1000} \times \frac{1}{3600} = 9.61 \times 10^{-5} \, \text{kWh}\)
1c: Heating water
Mass of water: 206.3 g
| Time (s) | Temperature (°C) | Energy (kWh) |
|---|---|---|
| 0 | 27.0 | 0.00 |
| 30 | 59.0 | 0.01 |
| 60.02 | 86.0 | 0.02 |
Mass of water: 406.2 g
| Time (s) | Temperature (°C) | Energy (kWh) |
|---|---|---|
| 0 | 34.0 | 0.00 |
| 10 | 34.0 | 0.01 |
| 38 | 46.0 | 0.02 |
| 69 | 64.0 | 0.03 |
| 101 | 84.0 | 0.04 |
1d: Brewing
Using 25g grounds = brew ratio of 16
Extraction time: 3.5 min
Part 2: Analysis
Heating Water
We begin with the electric kettle trials, which provide the clearest window into energy transfer behavior because water has a well-characterized \(C_p\). Two masses (206.3 g and 406.2 g) were heated, allowing us to examine how thermal mass affects both the rate of heating and total energy consumption.
As shown in Figure 1, both trials exhibit a roughly linear rise in temperature over time, consistent with the kettle delivering near-constant power throughout. The 206.3 g sample heats more steeply, reaching 86°C in about 60 seconds, while the 406.2 g sample climbs more gradually to 84°C after 101 seconds. Thus, more energy must be supplied to achieve the same temperature rise in a larger sample.
The cumulative energy plots in Figure 2 also show nearly linear trends and consistent with constant-power delivery. The larger mass consumes 0.04 kWh total compared to 0.02 kWh for the smaller, reflecting the greater energy requirement.
Plotting energy against temperature directly (Figure 3) again yields a nearly linear relationship for both trials, as expected from \(Q = mC_p\Delta T\): with \(m\) and \(C_p\) both approximately constant, energy and temperature rise proportionally. The two lines are offset because the larger mass requires more energy per degree of rise.
Calculating Heat Capacity
Using the overall \(\Delta T\) and total energy from each trial, we can back-calculate the apparent specific heat capacity of water. For the 206.3 g trial:
\[ \Delta T = 86.0 - 27.0 = 59.0^\circ C \]
\[ Q = 0.02 \,\text{kWh} \times 3.6 \times 10^6 \,\frac{\text{J}}{\text{kWh}} = 72{,}000 \,\text{J} \]
\[ C_p = \frac{Q}{m \Delta T} = \frac{72{,}000}{(206.3)(59.0)} = 5.91 \,\text{J/g·°C} \]
And for the 406.2 g trial:
\[ \Delta T = 84.0 - 34.0 = 50.0^\circ C \]
\[ Q = 0.04 \,\text{kWh} \times 3.6 \times 10^6 \,\frac{\text{J}}{\text{kWh}} = 144{,}000 \,\text{J} \]
\[ C_p = \frac{Q}{m \Delta T} = \frac{144{,}000}{(406.2)(50.0)} = 7.09 \,\text{J/g·°C} \]
| Mass (g) | ΔT (°C) | Total Energy (J) | \(C_p\) (J/g·°C) |
|---|---|---|---|
| 206.3 | 59.0 | 72,000 | 5.91 |
| 406.2 | 50.0 | 144,000 | 7.09 |
Both values are considerably higher than the accepted specific heat capacity of water of 4.18 J/g·°C, a 41.4% positive difference for the smaller trial and 69.6% for the larger. The dominant source of this discrepancy is the meter’s coarse 0.01 kWh resolution, which could be rounding measured energy upward. The true energy delivered to the water could be up to 0.01 kWh less than recorded in each case, meaning our \(C_p\) values are upper bounds rather than true measurements. Secondary contributions include heat lost to the surroundings and energy absorbed by the kettle body itself, both of which cause the meter to register more energy than actually entered the water.
Roasting
We next examine the roasting process. Unlike the kettle, the thermal camera here measures chamber air temperature rather than the beans directly, which underestimates the true temperature inside the roaster.
The temperature plot (Figure 4) shows a steady, roughly linear rise from 23°C to 89.2°C over 120 seconds. However, in the energy plot (Figure 5), energy accumulates in the first 60 seconds, then plateaus at 0.02 kWh through the end of the recorded window. This divergence of temperature continuing to climb while recorded energy stays flat is most likely a meter resolution effect. The actual energy draw during the second half of the roast was real but insufficient to push the cumulative total past the next 0.01 kWh threshold. Residual heat stored in the roasting chamber likely also contributed to the continued temperature rise after the element may have partially cycled off.
Comparing Energy Across Appliances
To put all three steps in perspective, we normalize energy consumption by mass — per gram of coffee for the roaster and grinder (40 g batch), and per gram of water for the kettle (406.2 g trial).
As Figure 6 illustrates, the roaster is by far the most energy-intensive step per gram of material processed at \(5.00 \times 10^{-4}\) kWh/g, roughly five times higher than the kettle at \(9.85 \times 10^{-5}\) kWh/g, and over 200 times higher than the grinder at \(2.40 \times 10^{-6}\) kWh/g! The grinder’s contribution is effectively negligible from an energy standpoint. The high energy expenditure of roasting (plus equipment) is likely what makes roasted coffee beans much pricier than green coffee beans at a large scale.
In absolute terms, the total energy consumed across all three appliances was:
\[ E_\text{total} = E_\text{roaster} + E_\text{grinder} + E_\text{kettle} = 0.02 + 9.61 \times 10^{-5} + 0.04 = 0.0601 \, \text{kWh} \]
At the local energy rate of $0.017/kWh, the total cost of the entire coffee-making process was:
\[ \text{Cost} = 0.0601 \, \text{kWh} \times \$0.017/\text{kWh} \approx \$0.00102 \]
The process cost approximately one-tenth of a cent in electricity. Even at such a low unit price, the roaster and kettle together account for over 99.9% of the expenditure.